Make a Positive Array

Make a Positive Array

What is this problem about?

Note: Due to overlapping naming conventions in programming platforms, "Make a Positive Array" generally refers to array transformation problems. Here we describe a common archetype associated with prefix constraints. You are given an array of integers. Your goal is to apply operations (like adding a constant X to all elements, or shifting values) such that every element in the array—or the running prefix sum of the array—remains strictly positive ($> 0$). You usually need to find the minimum initial value or minimum operations required to achieve this.

Why is this asked in interviews?

This problem archetype evaluates a candidate's understanding of Prefix Sums and identifying bottlenecks. Interviewers use it to see if a candidate can abstract a moving sequence of numbers into a single minimum value. It separates candidates who simulate the process step-by-step from those who can find a mathematical $O(N)$ shortcut.

Algorithmic pattern used

This problem heavily relies on the Prefix Sum / Greedy pattern. If you need to find a starting value X so that X + prefix_sum[i] > 0 at all times, you don't need to guess X. You simply calculate the standard running prefix sum starting from 0. As you iterate, you keep track of the absolute lowest dip the prefix sum takes. If the lowest dip is, for example, -4, then your starting value must be at least 5 to ensure it never drops to 0 or below.

Example explanation

Let the array representing changes be [-3, 2, -3, 4, 2]. We track the running sum starting from 0:

• Index 0 (-3): Sum = -3. Lowest dip = -3.
• Index 1 (2): Sum = -1. Lowest dip = -3.
• Index 2 (-3): Sum = -4. Lowest dip = -4.
• Index 3 (4): Sum = 0. Lowest dip = -4.
• Index 4 (2): Sum = 2. Lowest dip = -4.

The prefix sum dipped down to a minimum of -4. To ensure that the sum never drops to 0 or below at any point, we must offset this lowest dip. If we start with $X = 4$, the lowest point becomes 0 (not strictly positive). If we start with $X = 5$, the lowest point becomes 1. Therefore, the minimum starting positive value required is abs(-4) + 1 = 5.

Common mistakes candidates make

A frequent mistake is using a Binary Search or brute-force simulation to guess the starting value. While guessing $X = 1$, simulating the array, failing, then guessing $X = 2$ works, it is wildly inefficient ($O(N \log M)$ or worse). Another common error is just summing the negative numbers and ignoring the positive numbers; the sequence matters because a positive number might offset a later negative number!

Interview preparation tip

When an interview question asks you to find a "minimum starting value" to keep a sequence positive, immediately think of tracking the lowest point of a Prefix Sum. The entire problem boils down to 1 - min_prefix_sum (assuming min_prefix_sum is $\le 0$). This simple $O(N)$ trick makes you look like an algorithmic expert.